# Fourier Series Solved Problems (Then t-x = 0 which means the integral can be further simplified) According to my understanding, t is the time domain and any domain x can be ...

Tags: Business Planning Process StepsCoursework Vs CourseworkJetblue Case Study Swot AnalysisFundraising Business PlanFree Online Business Plans4 Thesis ConclusionPostgraduate By Coursework Or ResearchSin In Scarlet Letter Essay

It is instead done so that we can note that we did this integral back in the Fourier sine series section and so don’t need to redo it in this section.

Using the previous result we get, $= \frac\hspace\hspacen = 1,2,3, \ldots$ In this case the Fourier series is, $f\left( x \right) = \sum\limits_^\infty \sum\limits_^\infty = \sum\limits_^\infty$ If you go back and take a look at Example 1 in the Fourier sine series section, the same example we used to get the integral out of, you will see that in that example we were finding the Fourier sine series for $$f\left( x \right) = x$$ on $$- L \le x \le L$$.

$\begin & = \frac\int_^ = \frac\left[ \right]\\ & = \frac\left[ \right] = \frac\left[ \right] = L\end$ $\begin = \frac\int_^ & = \frac\left[ \right]\\ & = \frac\left[ \right]\end$ At this point it will probably be easier to do each of these individually. \right|_^0 = \frac\sin \left( \right) = 0\] $\begin\int_^ & = \left. \right|_0^L\\ & = \left( \right)\left( \right)\\ & = \left( \right)\left( \right)\end$ So, if we put all of this together we have, $\begin & = \frac\int_^ = \frac\left[ \right]\\ & = \frac\left( \right)\,\,\,,\hspace\hspacen = 1,2,3, \ldots \end$ So, we’ve gotten the coefficients for the cosines taken care of and now we need to take care of the coefficients for the sines.

$\begin = \frac\int_^ & = \frac\left[ \right]\\ & = \frac\left[ \right]\end$ As with the coefficients for the cosines will probably be easier to do each of these individually. \right|_^0 = \frac\left( \right) = \frac\left( \right)\] $\begin\int_^ & = \left. \right|_0^L\ & = \left( \right)\left( \right)\ & = \left( \right)\left( \right) = - \frac\end$ So, if we put all of this together we have, $\begin = \frac\int_^ & = \frac\left[ \right]\ & = \frac\left[ \right] = - \frac\left( \right)\hspace\hspacen = 1,2,3, \ldots \end$ So, after all that work the Fourier series is, $\beginf\left( x \right) & = \sum\limits_^\infty \sum\limits_^\infty \ & = \sum\limits_^\infty \sum\limits_^\infty \ & = L \sum\limits_^\infty - \sum\limits_^\infty \end$ As we saw in the previous example there is often quite a bit of work involved in computing the integrals involved here.

Due to the nature of the mathematics on this site it is best views in landscape mode.

If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Doing this gives, $\int_^ = \sum\limits_^\infty \sum\limits_^\infty$ We can now take advantage of the fact that the sines and cosines are mutually orthogonal.

The integral in the second series will always be zero and in the first series the integral will be zero if $$n \ne m$$ and so this reduces to, $\int_^ = \left\{ \right.$ Solving for  gives, $\begin& = \frac\int_^\ & = \frac\int_^\hspace\hspacem = 1,2,3, \ldots \end$ Now, do it all over again only this time multiply both sides by $$\sin \left( \right)$$, integrate both sides from –$$L$$ to $$L$$ and interchange the integral and summation to get, $\int_^ = \sum\limits_^\infty \sum\limits_^\infty$ In this case the integral in the first series will always be zero and the second will be zero if $$n \ne m$$ and so we get, $\int_^ = \left( L \right)$ Finally, solving for  gives, $= \frac\int_^\hspace\hspacem = 1,2,3, \ldots$ In the previous two sections we also took advantage of the fact that the integrand was even to give a second form of the coefficients in terms of an integral from 0 to $$L$$.

## Comments Fourier Series Solved Problems

• ###### First term in a Fourier series video Khan Academy
Reply

The first term in a Fourier series is the average value DC value of the function being approximated. Questions. Sal has simplified the giant equation above and arranged it to isolate and solve for a_0. He looks at the right side of the.…

• ###### E1.10 Fourier Series and Transforms - Department of.
Reply

Easier problem Complicated waveform → sum of sine waves. → linear. maths 1 lecture. Fourier series for periodic waveforms 4 lectures. Fourier. Example. Linearity. Summary. E1.10 Fourier Series and Transforms 2014-5379. Fourier.…

• ###### Fourier Series - University of Miami Physics
Reply

Fourier series started life as a method to solve problems about the flow of. The idea of Fourier series is that you can write a function as an infinite series of sines.…

• ###### Fourier Series introduction video Khan Academy
Reply

The Fourier Series allows us to model any arbitrary periodic signal with a combination of sines and cosines. In this video sequence Sal works out the Fourier Series of a square wave. Questions. because a lot of differential equations are easy to solve when you involve sines and cosines, but not as obvious to solve when.…

• ###### Solutions of Problems on Fourier Analysis of Continuous.
Reply

Solutions of Problems on Fourier. Analysis of Continuous Time Signals Unit 1 à. 4.1 Expansion of Periodic Signals by. Complex Exponentials the Fourier Series.…

• ###### Chapter 11 Fourier Series
Reply

Dec 13, 2013. Fourier Series is invented by Joseph Fourier, which basically asserts. Below, let's try to follow Fourier's steps in solving this problem and see.…

• ###### Introduction to Fourier Series
Reply

Oct 15, 2014. where a0, an, and bn are called the Fourier coefficients of fx. The following examples are just meant to give you an idea of. Example 1.…

• ###### Fourier series DE overview
Reply

OVERVIEW — SOLVING ODES WITH FOURIER SERIES. Read this in conjunction with the examples in class and homework. Section 9.3. We solve the.…

• ###### Fourier Series - Stewart Calculus
Reply

Fourier Series. When the French mathematician Joseph Fourier 1768–1830 was trying to solve a prob- lem in heat conduction, he needed to express a.…

• ###### Newest 'fourier-series' Questions - Mathematics Stack Exchange
Reply

Finding Fourier series for function ft=cos3t⋅sin5t in complex form. here entire series and fourier series hello guys i need to solve this problem before 8 hours.…